Let $\mathbf{n}$ is a Gaussian random vector with mean $\mathbf{0}$ and co-variance matrix $\mathbf{H}$. Let $\mathbf{r} = Sign(\mathbf{n})$, where $Sign(n_i) = 1$ if $n_i>0$ and $Sign(n_i) = -1$ if $n_i<0$. Let $\mathbf{A}$ be a positive semi definite matrix. Let $\sigma = \mathbf{r}^{T}\mathbf{A}\mathbf{r}$. I wanted to know if it is possible to get a concentration inequality for $\sigma$, like
$P(\sigma>E\{\mathbf{n}^{T}\mathbf{A}\mathbf{n}\})\leq$ some function, f(N) of the dimensions of vector $\mathbf{r}$.
Any references also would be very helpful. Thanks in advance!
We can note that $E\{\mathbf{n}^{T}\mathbf{A}\mathbf{n}\} = trace (\mathbf{AH})$, $E\{\mathbf{r}^{T}\mathbf{A}\mathbf{r}\} =\frac{2}{\pi}trace (\mathbf{A}\tilde{\mathbf{H}})$, where $\tilde{\mathbf{H}} = arcSin(\mathbf{H})$, where $arcSin$ (inverse sine) is taken over each entry of $\mathbf{H}$ and $trace(\mathbf{AH})\leq trace(\mathbf{A}\tilde{\mathbf{H}})$ and entries of $\mathbf{H}$ are in the interval $[-1,1]$.